Vector valued multivariate spectral multipliers, Littlewood–Paley functions, and Sobolev spaces in the Hermite setting (original) (raw)
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arXiv (Cornell University), 2013
In this paper we find new equivalent norms in L p (R n , B) by using multivariate Littlewood-Paley functions associated with Poisson semigroup for the Hermite operator, provided that B is a UMD Banach space with the property (α). We make use of γ-radonifying operators to get new equivalent norms that allow us to obtain L p (R n , B)-boundedness properties for (vector valued) multivariate spectral multipliers for Hermite operators. As application of this Hermite multiplier theorem we prove that the Banach valued Hermite Sobolev and potential spaces coincide. h k (x)f (x)dx, f ∈ L 2 (R n). According to [45, Lemma 1.2], the space C ∞ c (R n) of smooth compactly supported functions in R n is contained in the domain D(H) of H and Hf =Hf , f ∈ C ∞ c (R n). Harmonic analysis associated with Hermite polynomial expansions were begun by Muckenhoupt ([35] and [36]) who considered the one dimensional setting. Later, Sjögren ([42]), Fabes, Gutiérrez and Scotto ([17]) and Urbina ([53]) studied harmonic analysis operators associated
Journal of Functional Analysis, 2012
In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2 ((0, ∞), dt/t), γ(H, B) represents the space of γradonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BM O L (R n , B) (respectively, H 1 L (R n , B)) into BM O L (R n , γ(H, B)) (respectively, H 1 L (R n , γ(H, B))), where BM O L and H 1 L denote BM O and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BM O L (R n , B) and H 1 L (R n , B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.
arXiv (Cornell University), 2012
In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2 ((0, ∞), dt/t), γ(H, B) represents the space of γradonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BM O L (R n , B) (respectively, H 1 L (R n , B)) into BM O L (R n , γ(H, B)) (respectively, H 1 L (R n , γ(H, B))), where BM O L and H 1 L denote BM O and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BM O L (R n , B) and H 1 L (R n , B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.
Mixed norm estimates for Hermite multipliers
In this article mixed norm estimates are obtained for some integral operators, from which those for the Hermite semigroup and the Bochner Riesz means associated with the Hermite expansions are deduced. Also, mixed norm estimates for the Littlewood Paley g functions and g* functions for the Hermite expansions are obtained, which lead to those for Hermite multipliers
Besov and Triebel-Lizorkin Spaces Associated to Hermite Operators
Journal of Fourier Analysis and Applications, 2014
Let X be a space of homogeneous type and L be a nonnegative self-adjoint operator on L 2 (X) satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spacesḂ α,L p,q,w (X) and weighted Triebel-Lizorkin spacesḞ α,L p,q,w (X) associated with the operator L for the full range 0 < p, q ∞, α ∈ R and w being in the Muckenhoupt weight class A ∞. Under rather weak assumptions on L as stated above, we prove that our new spaces satisfy important features such as continuous characterizations in terms of square functions, atomic decompositions and the identifications with some well-known function spaces such as Hardytype spaces and Sobolev-type spaces. One of the highlights of our result is the characterization of these spaces via noncompactly supported functional calculus. An important by-product of this characterization is the characterization via the heat kernel for the full range of indices. Moreover, with extra assumptions on the operator L, we prove that the new function spaces associated with L coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of L, the spectral multiplier of L in our new function spaces and the dispersive estimates of wave equations.
A T1 criterion for Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space and applications
2010
In this paper we establish a T1 criterion for the boundedness of Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associated to the Hermite operator H. We apply this criterion in a systematic way to prove the boundedness on BMO_H(R^n) of certain harmonic analysis operators related to H (Riesz transforms, maximal operators, Littlewood-Paley g-functions and variation operators).
Square functions in the Hermite setting for functions with values in UMD spaces
Annali di Matematica Pura ed Applicata (1923 -), 2013
In this paper, we characterize the Lebesgue Bochner spaces L p (R n , B), 1 < p < ∞, by using Littlewood-Paley g-functions in the Hermite setting, provided that B is a UMD Banach space. We use γ-radonifying operators γ (H, B) where H = L 2 ((0, ∞), dt t). We also characterize the UMD Banach spaces in terms of L p (R n , B) − L p (R n , γ (H, B)) boundedness of Hermite Littlewood-Paley g-functions.
International Mathematics Research Notices, 2019
Let XXX be a space of homogeneous type and let LLL be a nonnegative self-adjoint operator on L2(X)L^2(X)L2(X) that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for LLL on the Besov and Triebel–Lizorkin spaces associated to LLL. Our work not only recovers the boundedness of the spectral multipliers on LpL^pLp spaces and Hardy spaces associated to LLL but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.
Journal of Fourier Analysis and Applications, 2016
Let X 1 and X 2 be metric spaces equipped with doubling measures and let L 1 and L 2 be nonnegative self-adjoint second-order operators acting on L 2 (X 1) and L 2 (X 2) respectively. We study multivariable spectral multipliers F(L 1 , L 2) acting on the Cartesian product of X 1 and X 2. Under the assumptions of the finite propagation speed property and Plancherel or Stein-Tomas restriction type estimates on the operators L 1 and L 2 , we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator F(L 1 , L 2) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space X 1 × X 2. We apply our results to the analysis of secondorder elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner-Riesz means. Contents 1. Introduction 1 2. Notation and preliminary results 6 2.1. Finite propagation speed property for the wave equation 8 2.2. Stein-Tomas restriction type estimates 9 3. The Hardy space H p L 1 ,L 2 (X 1 × X 2) 9 3.1. Atomic decomposition for the Hardy space H 1 L 1 ,L 2 (X 1 × X 2) 10 3.2. An interpolation theorem 12 4. Multivariable spectral multiplier theorems 12 5. Off-diagonal estimates for multivariable spectral multipliers 14 6. Proofs of Theorems 4.1 and 4.2 23 7. Applications 30 7.1. Riesz-transform-like operators 30 7.2. Double Bochner-Riesz means 31 References 31
Remarks on spectral multiplier theorems on Hardy spaces associated with semigroups of operators
2009
Let L be a non-negative, self-adjoint operator on L^2(\Omega), where (\Omega, d \mu) is a space of homogeneous type. Assume that the semigroup {T_t}_{t>0} generated by -L satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that f belongs to the Hardy space H^1_L if the square function S_h f(x)=(\iint_{\Gamma (x)} |t^2 L e^{-t^2 L} f(y)|^2 \frac{d\mu(y)}{\mu (B_d(x,t))} \frac{dt}{t})^{1/2}